2-factor theorem

In the <a href="/facts/Mathematical/pxTouaz4">mathematical</a> <a href="/facts/Discipline/1RItfSbh">discipline</a> of <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, the 2-factor theorem, discovered by <a href="/facts/Julius_Petersen/jdBwzHPH">Julius Petersen</a>, is one of the earliest works in graph theory. It can be stated as follows:

<blockquote>
Let 
 
 
 
 G
 
 
 {\displaystyle G}
 
 be a <a href="/facts/Regular_graph/j9dSoLE6">regular graph</a> whose <a href="/facts/Degree_(graph_theory)/LnFFpI8v">degree</a> is an even number, 
 
 
 
 2
 k
 
 
 {\displaystyle 2k}
 
. Then the edges of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 can be partitioned into 
 
 
 
 k
 
 
 {\displaystyle k}
 
 edge-disjoint 2-factors.

</blockquote>
Here, a 2-factor is a subgraph of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 in which all vertices have degree two; that is, it is a collection of cycles that together touch each vertex exactly once.

2-factor theorem open-in-new

2-factor theorem